# End (topology)

From Knowino

In general topology, an **end** of a topological space generalises the notion of "point at infinity" of the real line or plane.

An end of a topological space *X* is a function *e* which assigns to each compact set *K* in *X* some connected component with non-compact closure *e*(*K*) of the complement *X* – *K* in a compatible way, so that

If *X* is compact, then there are no ends.

## Examples

- The real line
**R**has two ends, which may be denoted ±∞. If*K*is a compact subset of**R**then by the Heine-Borel theorem*K*is closed and bounded. There are two unbounded components of**R**–*K*: if*K*is contained in the interval [*a*,*b*], they are the components containing (-∞,*a*) and (*b*,+∞). An end is a consistent choice of the left- or the right-hand component. - The real plane
**R**^{2}has one end, ∞. If*K*is a compact, hence closed and bounded, subset of the plane, contained in a disk, then there is a single unbounded component of**R**^{2}–*K*, the component containing the complement of the disc.

## Compactification

Denote the set of ends of *X* by *E*(*X*) and let . We may topologise *X*^{ * } by taking as neighbourhoods of an end *e* the sets for compact *K* in *X*.

Some content on this page may previously have appeared on Citizendium. |